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Mathematics

¨  What is the importance of geometry in life ?
¨  Could I please get some help on converting kilograms into m2?
¨
  How would music be affected if the contributions of mathematics were not used?
¨  How long does it take to fill a pond?
¨  How has calculus changed the world?
¨  I like building things but don't like math.  Should I go into engineering school?
¨  Who invented mathematics?
¨  Who invented numbers?
¨  How far is the horizon?
¨  How do I figure out Earth's circumference at 41° latitude?

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QUESTION:
What is the importance of geometry in life?

ANSWER FROM Stephanie Wong on 22 April 2007:
Because we live in a multi-dimensional world, geometry is very important!  Everything that we see has a length, perimeter area and volume.  While it is not always important to know exactly how one geometrical parameter relates to something else (say, how does the length of the sides of a cube give its volume), it can be very useful in everyday life.  For example, what if you wanted to ship something in a box and the cost of the shipping depended on that?  Which is the best box to fit your item?  Suppose you are building a shed and you need to cut the lumber.  How do you know what angle cut to make?  Suppose you have a ladder and you want to see if it is long enough to reach a high window?  It would be good to know that, considering a safe angle to tilt the ladder, would it reach a high enough height?  Doing all these things by trial and error is possible, but having a trusty calculator and a simple mathematical formula would be much simpler.

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QUESTION:
Could I please get some help on converting kilograms into m2.  I have the weight of some steel plate and am trying to convert it to Square Metres.  Is there a formula I could put into an Excel spreadsheet?

ANSWER from Stephanie Wong on 16 September 2006:
Are you sure you want to convert from kilograms to square metres?  That's because a kilogram quantifies MASS and a square meter quantifies AREA.  They are incompatible.

However, if what you want to do is to determine the size of your square plate from the mass of it, then that's a different matter.  But you will need to know the density (kg/m^3) and thickness of the steel (m) you're working with.

Density D=mass/volume

To get volume, V=m/D

Divide the volume by the thickness to get the area of a plate:
area = V/thickness (the answer is in m^2)

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QUESTION:
How would music be affected if the contributions of mathematics were not used?

ANSWER from Stephanie Wong on 24 February 2006:
It's hard to say because music HAS been affected by mathematics.  While coming up with a tune does not require music, people who have analyzed music for its mathematical patterns and who then had clout to express their opinions on music or were to compose based on their findings have undoubtedly changed music.  This is quite a large subject but I will give you a few examples.

The Ancient Greek philosophers were quite critical of music.  Note that the Greeks classified music as "harmonics", which was a subgroup of the "quadrivium" (arithmetic, geometry, astronomy, and harmonics).  I guess in today's terms we'd call that grouping "the sciences".  What we would consider "the arts" was in a group called the "trivium" (grammar, logic, and rhetoric).  So, it is interesting that the Greeks thought music was a science instead of an art.  Given that, Greek music was based on "modes", something like our current musical scale system.  Because of the inherent properties in each mode, music could influence the soul in different ways.  Therefore, the choice of music was heavily affected by the mode, which the philosophers analyzed mathematically.  Also, it is believed that Pythagoras was the first to discover the concept of intervals with respect to numerical ratios.  Any musician knows how important intervals are.

On we go to the Medieval Europe.  In the Middle Ages, life was largely controlled by the Church.  Thus, music was heavily controlled by them as well.  A lot of music back then was considered unchristian, with these assumptions based on the teachings of the Ancient Greeks.  Medieval music sounds much different from modern classical music because of the intervals.  The medieval composers did not use much the interval of a 3rd which gives a characteristic sound.  One big thing back then is the essential banning of the "tritone".  This was a augmented 4th/diminished 5th interval.  If you try to play those two notes together now, it does sound pretty horrible.  But the Church back then pretty much banned all use of this interval because it "came from the devil".

When technology improved so that instrument-makers could fine-tune their instruments to a proper tuning, they had to consider the spacing (numerical pitch) between each of the notes.  A scale can never be tuned perfectly.  An instrument-maker has to tune an instrument a certain way to achieve the desired progression of pitches.

Nowadays, there is a lot of technology used in the creation of music.  Undoubtedly, this requires a lot of math.

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QUESTION:
How long does it take for the average newly built pond to fill in with water?  Is there a reasonably simple equation to estimate the time required?  Pond in question is 5 acres with inflow stream equal to the average bathtub spigot and the soil is sand-clay with an adequate dam including a clay core and average depth is 6 feet and location is Midlands region of South Carolina (Lexington County).

ANSWER from Roger Herzler on 23 January 2006:
Unfortunately, I believe you should ask this question of a local water or soils enginner.  There are a number of factors that will affect your question including:

- Evaporation rate of your local weather (highly variable)
- Seepage rate into the surrounding soils and/or aquifers
- The exact flow of water coming in (I don't know what a gallons per minute is of the 'average bathtub spigot')

This equation of how much water will be needed to fill a given volume might help: Volume = Length x Width x Depth (average) found at http://www.recreonics.com/fyi/calculating_pool_capacity.htm

With that equation you might come up with a reasonable gallons figure and can work up a back-of-the-envelope figure.  I worked a number with 43560 square feet (one acre) x 6 feet deep and came up with 261360 cubic feet of water per acre.  Then multiply that by 5 and I get 1306800 feet^3 (cubic feet) of water.  Lastly, from that figure we can get gallons with 1306800 x 7.481 or 9,776,170.8 gallons.  That conversion equation comes from: http://www.wrightwater.com/wwe/wwewaterquantity.htm

However, if this is a real 5 acre pond (not just a thought exercise), you're talking about serious water storage volumes that can affect your surrounding area if your dam, etc. were to fail.  Hence the suggestion you ask a qualified engineer in your area to get a better set of figures.

I hope that gives you a little direction to proceed in.

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QUESTION:
How has calculus changed the world?

ANSWER from Stephanie Wong on 27 November 2005:
There are two fundamental principles in calculus, the rate of change and the cumulative sum of small changes.  Prior to the development of calculus by Issac Newton and Gottfried Leibniz, the main fields of mathematics were algebra and geometry.  Algebra is very useful in quantifying, giving you discrete answers for static systems, or at its finest, describing and explaining the pure atomic structure of what is math through the basic axioms.  Geometry gives us the shape of the world, and all the visual (and invisible) structures and patterns known to man.

However, we haven't addressed change.  How does something "change"?  The equations of a static construct are fine, but what if the construct changes?  If a quantity abruptly changed once, you could easily recalculate, but when a quantity is continuously changing, we must find a way to quantify the change mathematically instead of treating each quantity at an instantaneous moment.  This is where calculus comes in.  At its simplest, we may deal with a straight line on a graph.  A plot of y versus x which gives us a straight line.  You might know that taking the final (x,y) and subtracting from it the initial (x,y) that you get something called the slope.  What is the slope?  It is a rate of change.  A good example is the speed of a car at time "A" compared to the speed at time "B".  Not all things are so simple and "linear", however.  When you begin to work with curves, or with changes to the rate of a change, you will want to work with calculus.  Think about all the things in the world that involve "change", and you can see how calculus is important in the world.

The second part of calculus is integral calculus.  This involves the sum of contributions over a period of change.  A simple example would be the amount of money a child made each hour at his/her lemonade stand.  One can make a bar graph for each hour, with the height of each bar giving the amount of money made.  To get the cumulative sum of earnings, you can add up each bar height and multiply by the number of hours he/she worked.  That is a crude integration, and is actually the fundamental principle of integral calculus, the Riemann sum.  What if you recorded the earnings per 10 minutes?  You could make a graph of that.  What if you were messy and picked different time intervals?  Suppose the child knew that the amount of sales depended on the temperature outside and the time of day.  He/she could guess how much lemonade could be sold during each of these conditions.  If the child was very bright, he/she could derive a formula that would indicate that given these conditions, the stand would earn this much in this period of time.  Given today's parameters, the child could figure out how much money could be made over the whole day.  Soon, this would be very difficult to figure out using a bar graph.  But coming up with the right "equation", and then applying basic integration, the child could figure out how much money would be made.

I hope these two simplistic examples give you a flavour of WHAT calculus is.  Now how has it changed the world?  Look at the things around you.  What involves rates of change and what involves the accumulation of change?  It is found in every facet of society.  Calculus is not used only in science, but is important in economics, sports and even music, among other things.  Begin to research calculus, if you haven't already, by learning the basic principles of calculus.  Read up on how it was conceived and more importantly, WHY it was conceived.  Then, depending on your interests, pick a topic and follow through to see how calculus has affected the invention and development of this procedure/object.  Investigate why calculus is such a mainstream course in university?  Is there a rhyme or reason to it?  When you begin to think about it, you may see that calculus is not as esoteric as it may seem.  It has revolutionized human society.

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QUESTION:
I really enjoy the building aspect of engineering and have built many "toys".  I have received my BS degree and have the opportunity to stay in school to earn an engineering degree but I am not sure that I want to go through 3 more years of engineering school (math) to get that degree.  I would appreciate any advice you can give me.

ANSWER from Roger Herzler on 2 September 2005:
This is a tough question to answer, and I'll probably give similar advice to your local college career assessment specialist.  I think your core question should be: "What type of career will make me happy in life?"  Not only do engineers have to spend a lot of time in school doing math and taking math classes, but they do a lot of it in day-to-day work as well.  It may not always be vector analysis, but it isn't unheard of (and probably should be expected) to be performing calculus on a daily basis in the course of a normal day for example.  Yes, engineers get to spend a lot of time thinking outside the box like you did building various "toys" around your farm, but you can't get away from the math if the devices you're constructing are truly going to work their best and most efficient - something usually demanded by employers. :o)

What you may find when you're forced to do math on a daily basis in your classes is that you truly enjoy the heavier mathematics engineers have to take.  However, you may not.  My advice would be to seriously evaluate what you think will make you happy, because if you've got that in life you're going to be excited to go to work, passionate about what you do, and you'll likely do the very best possible.  That's a win for everyone involved.

ANSWER from Homer Hickam on 3 September 2005:
If you don't want to keep going to school to get your engineering degree, then you probably shouldn't do it.  Let that space be taken by someone who really wants and needs the education.  You already have a degree in an associated field.  Use that and go ahead into the workplace to make your way.  You might consider military service.  Certainly, our country needs young men to volunteer their time and skills.  That's what I did when I graduated from college and it was the best thing I could have possibly done.  I learned a great deal during those years in the military and look back on them with great fondness.  Good luck.

ANSWER from Michael Bastoni on 14 September 2005:
Engineering without any math is a Sisyphian task.  Simply put; Engineering without math would require ceaseless, futile and hopeless labor.

Note on the (above) reference to Sisyphus:

"The gods had condemned Sisyphus to ceaselessly rolling a rock to the top of a mountain, whence the stone would fall back of its own weight.  They had thought with some reason that there is no more dreadful punishment than futile and hopeless labor." - Albert Camus

Engineering without the application of math and science can be done... but the trade off is time and personal energy and the acceptance of an inordinately high degree of failure... almost unendurable failure, a Sisyphian task for certain.

There once lived a man, more an inventor than an engineer, who was able to successfully make that trade off... there are few in the history of technology who were more tenacious, more demanding of themselves and of others, few who worked more tirelessly to achieve the means they sought, few more driven to understand the nature of things through direct observation and experimentation... few more willing to ceaselessly roll the rock up the hill, few like Thomas A. Edison.

If you want to be inspired by an engineering story of an accomplished technical mind that used little if any applied math... read Edison a Biography by Matthew Josephson.  The story is as inspiring as it frustrating.

With the application of even a little math, Edison could have worked so less hard, failed so less often and accomplished even more discoveries.

Why do engineers use math anyway?

Engineers use math skills for several reasons:

1. Engineers design, build and maintain things.  The things they design, build and/or maintain are made of materials and components.  Making informed decisions about the materials and components used in the things they design, build and/or maintain, requires the application of math and physics fundamentals.

I find myself speeding down the highway, or flying over cities and thinking how grateful I am that the men and women who designed, built and or maintained the parts used in the planes and cars I travel in, were able to do so with mathematical certainty... since in the absence of math, they could have only made guesses about the strength, high temperature and stress performance of those mechanical components that protect my life and safety... and yours.

This is why mathematical proficiency is a requirement for obtaining an engineering degree.

2. To help them with modeling and simulations to predict the behavior of mechanisms before they are actually built.

For instance: This year my students and I are designing and building a wind turbine that will produce 1KW (Average) of electrical power.

The first thing we did was ask the question, "How much energy (power) is in the wind?"

We derived a mathematical expression using the kinetic energy formula KE=1/2 * Mass * (Velocity)^2

We then built a spread sheet that would allow us to determine the raw power of the wind for any wind speed and turbine area... and then we dialed in Betz' law (efficiency rule) and we had a simple tool that allowed us to quickly and easily determine the necessary size of our turbine.  We could effectively use this spread sheet to build innumerable wind turbines.  We did all this from scratch in about 3 hours including the necessary research.

Now it would have taken about 5 weeks of constant effort to shape a turbine, erect it and then test it to see how much power it put out... if it was substantially more or less we would have had to repeat the process... this is much like how Edison worked.

We like using math better.

35 years ago things were different... I did not like math as a high school student.  I did not appreciate then what a helpful tool it was.  I did like to build things... custom cars, choppers and even an old 57' school bus turned camper which we used to tour the US and Canada... but I had no use for math.

The problem was that I could not appreciate the value of math.  I did not know what it could do for me.  I am still not a great mathematician... I will never be.  But I realize that the effort required to use math as a tool... far, far, far outweighs the effort required to accomplish any technical endeavor without using math.  I learned this when I started my first construction company.  We built custom homes in the Virgin Islands.

Beautiful exposed beam ceilings are what trigonometry looks like when you make it out of wood.  Large clear spans are possible when you can calculate stress and strain limits of wood fibers and the bending moments of steel girders.  Comfortable interior climate control is only possible through the application of heat loss studies and the selection of appropriate building materials.  Moreover, a contractor who cannot make mathematical determinations of building costs BEFORE the building is built will soon go broke or more likely end up in a costly court battle... and so I came to appreciate the power of math and I was surprised how quickly I learned to use it.

This will happen to you if you pursue your passion.  You will learn to love math for what it can do for you and how it will enhance the creative experience of designing and building things.  Engineering is a team sport.  There are many "positions" on the engineering team.  There are engineering and technical positions that only require a basic understanding and application of geometry, algebra and trigonometry.  Each person on that team can contribute to the engineering effort.

My son is not a gifted mathematician but he loves machines and he loves to travel.  He is now studying for a degree in marine engineering.  He will run the big ships and travel the globe.  He will have to struggle through calculus, but he will get help.  He will learn what he needs to know so that he will be able to get the right answers to the right questions and he will learn to recognize the right answers from the wrong ones.  It won't be easy, but then, engineering without math is even harder.  It won't be easy for him but it will be possible.  I suspect the same is true for you.

My advice... continue to be creative in your technical pursuits.  Build things.  Take things apart and in doing this you will build a personal library of empirical knowledge.  A kind of knowledge that many AP math students lack... and then start using math to analyze and model the performance of the things you design and build.

Some people go to gyms and work out to improve their physical condition... you can employ the help of a personal math trainer... a tutor, and work out in the gymnasium of the mind to improve your math condition.  You can do it, you will come to enjoy it.

Engineering is hard work... with or without math.  You can spend 3 hard years pursuing your degree or you can spend a lifetime hammering out the answers to knotty mechanical problems that could have been solved quickly and easily with a pencil and paper, your choice.  Either way requires effort.  But the degree is a form of validation that supports the experience and knowledge that follows.

Read Edison... then decide.  I suspect you'll work for the degree... in retrospect, even Edison would have to agree... and he would not do so willingly.

I hope you find this story useful.  No matter what you decide, the choice is yours and yours alone.

My most often offered advice is this: When given a difficult choice where either option seems right, one asks "Which is the best decision?"

I often answer... "The best decision is most likely the one perceived to be the most difficult."

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QUESTION:
Who invented mathematics?  In other words, what is the history of mathematics?

ANSWER from Yoonjung Huh on 15 September 2005:
Like other studies (music, visual arts, science, literature, etc.), mathematics was not invented by a person or two.  For example, people were interested in counting objects and constructing useful tools, which naturally lead to the beginning of algebra and geometry, respectively.  It certainly has rich history, and the following sites may be helpful:
http://www-groups.dcs.st-and.ac.uk/~history/
http://www.cln.org/themes/math_history.html
You can also find some information on numbers in our Q & A Archives:
http://www.imagiverse.org/questions/archives/

I recommend the following book, Fermat's Enigma by Simon Singh - it's about the recently (in the 1990's) solved problem that mathematicians had worked on for many many years.  It covers a good amount of history of mathematics.  If you would like to read a bit more advanced material, Journey through Genius by William Dunham is a good book to read.

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QUESTION:
Who invented numbers?  Who discovered numbers?

ANSWER from Stephanie Wong on 30 August 2005:
Probably back in the Stone Age, humans realized that items around them could be categorized as none, a little, and a lot.  Then, they realized that they could quantify these items, as 1 item, 2 items, 3 items, etc.  Quickly, they used their own fingers to count.  When humans developed writing, they used tick marks to denote amounts, and soon they developed special symbols to make what we called numbers.

The numeric system that we use today was developed in India around 500 AD.  It is a positional system with a base of 10.  That is, numbers were ordered and could be expressed in powers of ten.  For example, if you just used tick marks to count, that would not be positional, and it would probably be very difficult to do math in!  Another positional system but with base 2 are the binary numbers.

The origin of "zero" is more elusive.  It might seem obvious to us to have zero as a number (denoting nothing) or as a place value.  However, it took some time for this concept to develop.

The modern symbols, 0 through 9, were developed about 1000 AD by the Arabs.

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QUESTION:
If a person who is 1.8 meters tall is standing on a very flat land, how far is the horizon from where this person is standing?

ANSWER from Stephanie Wong on 21 August 2005:
In response to your question about the distance to the horizon, I used some simple trigonometry to solve for it.  If you follow the steps that I have given, you can easily calculate the answer.  Please see my article describing it:

How Far is the Horizon?
http://www.imagiverse.org/resources/howthingswork/horizon/

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QUESTION:
How do I figure out Earth's circumference at 41 degrees latitude?

ANSWER from Stephanie Wong on 7 December 2003:
To calculate the circumference of a circle defined by the 41° latitude line, we need to find out the radius of this circle.  It can be simply done when knowing the radius of the Earth, which is about R=6370 km.

Draw a circle and label its radius along the horizontal (0°).  Draw another radius line so that it forms a 41° angle from the horizontal.  We now have an acute angle inside this circle with angle 41°.  The inclined line intersects the "Earth" at a latitude of 41°, call that point Y.  Draw a line from Y that perpendicularly intersects the horizontal line, call that point X.  The length of the line from the center of the circle to point X is our desired radius at 41°, call that x.  You can now see that we have made a right triangle so that we may use trigonometry.

Our inclined radius R is our hypotenuse.  Let x be the adjacent side.

The correct trigonometric formula to use is:

cos(t) = adj/hyp , where t is the angle in degrees.

So, cos(41°) = x/R

Upon rearranging,

x = Rcos(41°)

And you can easily solve for X when you plug in R.

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